model category, model $\infty$-category
Definitions
Morphisms
Universal constructions
Refinements
Producing new model structures
Presentation of $(\infty,1)$-categories
Model structures
for $\infty$-groupoids
on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
for equivariant $\infty$-groupoids
for rational $\infty$-groupoids
for rational equivariant $\infty$-groupoids
for $n$-groupoids
for $\infty$-groups
for $\infty$-algebras
general $\infty$-algebras
specific $\infty$-algebras
for stable/spectrum objects
for $(\infty,1)$-categories
for stable $(\infty,1)$-categories
for $(\infty,1)$-operads
for $(n,r)$-categories
for $(\infty,1)$-sheaves / $\infty$-stacks
An object in a model category is bifibrant if it is both fibrant as well as cofibrant.
To some extent a model category-structure on a homotopical category/relative category may be understood as a device for finding/forming the full subcategory of bifibrant objects and with it a convenient presentation of the localization at the given class of weak equivalences:
Equipped with the homotopy equivalence-classes of its morphisms, the full subcategory of bifibrant objects is equivalently the homotopy category, hence the localization of the original category at the given class of weak equivalences, see here.
Better yet, if the model structure is sSet-enriched and otherwise good enough (for instance: combinatorial), then the full sSet-enriched category on the bifibrant objects is equivalently the $(\infty,1)$-category which is the simplicial localization at the given class of weak equivalences, see here.
This makes bifibrant objects a convenient notion for abstract reasoning about (simplicial) localizations.
On the other hand, in most model categories the bifibrant objects are not the ones that are conveniently handled in practice (often they can be produced only by abstract fibrant+cofibrant resolution-machines which tend to produce unwieldy results). But the tools provided by model category-structure also allow to reason about bifibrant objects without necessarily constructing them. For example, a key lemma says that for computing homotopy classes of maps between bifibrant objects it is actually sufficient to use an equivalent cofibrant object for the domain and an equivalent fibrant object for the codomain.
See at model category.
Created on May 14, 2023 at 12:49:46. See the history of this page for a list of all contributions to it.